I am confused on what is meant by the modulus of a Riemann surface. I have done some very quick searches through my books but haven't found anything.(adsbygoogle = window.adsbygoogle || []).push({});

For context I will paraphrase a portion of Penrose's road to reality that I'm reading.

In the complex plane construct a parallelogram by connecting cyclically the points ##0, 1, 1+p, p## for some complex number ##p##. Orient the region by ##0\rightarrow 1, p\rightarrow1+p, 0\rightarrow p, 1\rightarrow 1+p##. Form the quotient space by equating opposite sides. Here's the construction I'm looking at

https://www.dropbox.com/sh/h8ws4v10xn7s5bx/j1jxi603FI#f:modulus.PNG

URL: image

Hence this is a construction of a torus. The text then goes on to say "for differing values of ##p##, the resulting surfaces are generallyinequivalentto each other; that is to say, it is not possible to transform one into another by means of a holomorphic mapping."

I don't really see this. Suppose I choose a different ##p## that preserves the same orientation (say ##p+1## if ##p## is in the 1st quadrant). The parallelogram is different, but by the Riemann mapping theorem there's clearly a holomorphism between the two regions, and so the resulting Riemann surface should be equivalent, though the text says otherwise.

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# Modulus of Riemann Surface

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